On second κ -variation

• Autorzy: Ereú J. J. , Lopez L. M. , Merentes N. J.

Identyfikatory

DOI

10.14708/cm.v56i2.1233

• Języki publikacji: EN

Abstrakty

EN

We present the notion of bounded second κ-variation for real functions defined on an interval [a,b]. We introduce the class κBV²([a,b]) of all functions of bounded second κ-variation on [a,b]. We show several properties of this class and present a sufficient condition under which a composition operator acts between these spaces.

Słowa kluczowe

EN

functions of bounded second variation; functions of bounded variation; κ-function

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2016
• Tom: Vol 56, No. 2
• Strony: 209--224
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Ereú J. J.

• Departamento de Matemáticas, Decanato de Ciencias y Tecnología, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela

autor

Lopez L. M.

• Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela

autor

Merentes N. J.

• Escuela de Matemáticas, Universidad Central de Venezuela, Caracas, Venezuela

Bibliografia

• [1] J. Appell, J. Banaś, and N. Merentes, Bounded Variation and Around, De Gruyter Studies in Nonlinear Analysis and Applications, vol. 17, Berlin 2014.
• [2] J. Appell and N. Merentes, Composing/unctions 0/bounded Korenblum variation, Dynam. Systems Appl. 22 (2013), 197-206.
• [3] W. Aziz, J. A. Guerrero, J. L. Sanchez, and M. Sanoja, Lipschitzian composition operator in the space KBV [a, b], J. Math. Control Sci. Appl. 4 (2011), no. 1, 67-73.
• [4] D. Bugajewska, D. Bugajewski, and G. Lewicki, On nonlinear integral equations in the space of functions of bounded generalized f-variation, J. Integral Equations Appl. 21 (2009), no. 1, 1-20, DOI 10.1216/JIE-2009-21-1-1.
• [5] V. V. Chistyakov, On mappings of bounded variation, J. Dynam. Control Systems 3 (1997), no. 2, 261-289, DOI 10.1007/BF02465896.
• [6] V. V. Chistyakov, Mappings of generalized variation and composition operators, J. Math. Sci. (New York) 110 (2002), 2455-2466, DOI 10.1023/A:1015018310969.
• [7] D. S. Cyphert and J. A. Kelingos, The decomposition of functions of bounded K-variation into differences of K-decreasing functions, Studia Math. 81 (1985), 185-195.
• [8] P. L. Dirichlet, Sur la convergence des séries trigonemétriques que servent à représenter une function arbitraire entre des limites donnés, J. Reine Angew. Math. 4 (1826), 157-159.
• [9] J. Giménez, L. López, and N. Merentes, The Nemitskij operator on Lip(k)-type and AC(k)-type spaces, Demonstr. Math. 46 (2013), no. 3, 28-37.
• [10] J. Giménez, L. López, and N. Merentes, A Burenkov’s type result for functions of bounded K-variation, Ann. Funct. Anal. 6 (2015), no. 1,1-11, DOI 10.15352/afa/06-1-1.
• [11] C. Jordan, Sur la Série de Fourier, C. R. Acad. Sci. Paris 2 (1881), 228-230.
• [12] M. Josephy, Composing functions of bounded variation, Proc. Amer. Math. Soc. 83 (1981), no. 2, 354-356, DOI 10.2307/2043527.
• [13] S. K. Kim and J. Yoon, Riemman-Stieltjes integral of functions of K-bounded variation, Comm. Korean Math. Soc. 5 (1990), no. 2, 65-73.
• [14] B. Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), 187-219, DOI 10.1007/BF02392019.
• [15] B. Korenblum, A generalization of two classical convergence tests for Fourier series, and some new Banach spaces of functions, Bull. Amer. Math. Soc. 9 (1983), no. 2, 215-218, DOI 10.1090/S0273-0979-1983-15160-1.
• [16] N. Merentes, On functions of bounded (p, 2)-variation, Collect. Math. 43 (1992), no. 2,117-123.
• [17] N. Merentes and S. Rivas, El Operador de Composición en Espacios de Funciones con algún tipo de Variación Acotada, IX Escuela Venezolana de Matemáticas, Facultad de Ciencias-ULA, Mérida-Venezuela 1996.
• [18] F. Riesz, Sur certains systèmes singuliers d’équations intégrales, Ann. Sci. École Norm. Sup. (3) 28 (1911), 33-62.
• [19] A. W. Roberts and D. E. Varberg, Functions of bounded convexity, Bull. Amer. Math. Soc. 75 (1969), no. 3, 568-572, DOI 10.1090/S0002-9904-1969-12244-5.
• [20] A. M. Russell and C. J. F. Upton, A generalization of a theorem by F. Riesz, Anal. Math. 9 (1983), 69-77.
• [21] M. Schramm, Functions of O-bounded variation and Riemann-Stieltjes Integration, Trans. Amer. Math. Soc. 287 (1985), 49-63, DOI 10.2307/2000397.
• [22] D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107-117; Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity. II.
• [23] N. Wiener, The quadratic variation of a function and its Fourier coefficients, Z. Angew. Math. Phys. 3 (1924), 72-94.
• [24] L. C. Young, Sur une généralisation de la notion de variation de puissance p-ième bornée au sens de M. Wiener, et sur la convergence des séries de Fourier, C. R. Acad. Sci. Paris, Ser. A-B 204 (1937), 470-472.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).